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Unformatted text preview: “TAKE HOME” PORTION OF THE FINAL EXAM, DUE ON
EXAM DAY (1) Let Rd be the (inﬁnite) poset, whose ground set is the set of dtuples of
real numbers, and (a1 , . . . , ad ) ≤ (b1 , . . . , bd ) if and only if ai ≤ bi for all
i = 1, . . . , d. Prove that the dimension of a poset P is equal to the least d
such that P can be embedded1 into Rd .
(2) Let X = {a, b, c, d, e, f } be the ground set of a poset. Deﬁne the (strict,
i.e. irreﬂexive) relation to be {(a, b), (a, c), (a, d), (b, d), (e, c), (e, d), (e, f ), (f, d)}.
Show that the resulting poset is of dimension 3.
(3) Show that for n ≥ 3, if you remove a point from Sn , the resulting poset is
of dimension n − 1. 1An embedding of a poset A into a poset B is an injective orderpreserving map f : A → B .
Orderpreserving means that x < y in A if and only if f (x) < f (y ) in B .
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.
 Fall '09
 WILDSTROM
 Real Numbers

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